nLab
cotangent bundle

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a manifold (or generalized smooth space) XX, the cotangent bundle T *(X)T^*(X) of XX is the vector bundle over XX dual to the tangent bundle T *(X)T_*(X) of XX. A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 11-form.)

Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ω,v\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be dual to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ω,v\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

d af,v=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

df,v=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if fgf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω= iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x 1,,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.

Revised on September 1, 2017 07:01:12 by Urs Schreiber (192.41.135.223)